Stellar convective stability

Stellar convective stability compares the density of an adiabatically displaced fluid element with its new surroundings. Temperature and composition gradients determine whether buoyancy restores the element or amplifies its displacement.

In a thin ideal-gas grey atmosphere with $T^4=T_e^4(1/2+3\tau /4)$, constant enclosed mass $M$ and luminosity $L$, and $\kappa ={\kappa }_0\rho T^{13}$, divide the stellar hydrostatic equation by radiative diffusion in a star. This gives $PdP/dT=16\pi acGM\mathcal{R}{T}^{-9}/(3{\kappa }_{0}L\mu )$. Integrating from $P=0$, $T^4=T_e^4/2$, yieldsLogarithmic derivative gives $\nabla =T^8/T_e^8-1/4$. For a monatomic gas gas, the Schwarzschild criterion reaches equality at $\nabla =2/5$, hence $T/T_e=(13/20{)}^{1/8}$. The purely radiative profile cannot be continued past this point as a convectively stable model.

Stellar convective instability occurs when an adiabatic process displaces a fluid element and buoyancy amplifies the displacement. For uniform composition the Schwarzschild criterion gives instability when the actual temperature gradient exceeds the adiabatic temperature gradient; composition gradients instead require the Ledoux criterion.

Consider a spherical uniform-density stellar model of mass density $\rho$, radius $R$, negligible surface pressure and a fixed-composition ideal gas with adiabatic index $\gamma >1$. Define ${\omega }_d^2=4\pi G\rho /3$, so the inward gravitational acceleration has magnitude $g(r)={\omega }_d^{2}r$ and $P(r)=\rho {\omega }_d^2(R^2-r^2)/2$.

For an adiabatically displaced parcel in pressure balance with its surroundings, the squared buoyancy frequency isA constant background mass density does not mean an incompressible parcel: its pressure and temperature change during the displacement. The negative $N^2$ means a rising parcel is lighter than its surroundings and continues to rise. Equivalently, the ideal-gas specific entropy decreases outward.

Thus this thermally stratified equilibrium has stellar convective instability even if a separate test of homologous adiabatic stellar stability finds restoring radial compression for $\gamma >4/3$. Mechanical equilibrium, radial compression stability and buoyant stability answer different questions. The singular near-surface expression also reflects the zero-pressure idealized boundary and should not be extrapolated to a realistic photosphere unchanged.

For uniform composition, a stellar layer is convectively stable when $\nabla <{\nabla }_{\mathrm{ad}}$ and unstable when $\nabla >{\nabla }_{\mathrm{ad}}$, where $\nabla =d\log T/d\log P$.

The Ledoux criterion tests local adiabatic stellar convective stability in the presence of a composition gradient. Definewhere $\mu$ is the mean molecular weight and the partial derivatives describe the equation of state. For positive $\delta$, the stratification is stable to an infinitesimal composition-preserving adiabatic process displacement whenThe parcel adjusts its pressure to its surroundings, retains its composition and changes temperature along the adiabatic temperature gradient. Comparing its resulting mass density with the ambient value determines the restoring buoyancy. For positive $\varphi /\delta$, increasing mean molecular weight inward gives ${\nabla }_{\mu }>0$ and an additional stabilizing contribution. With uniform composition the condition reduces to the Schwarzschild criterion. Thermal or compositional diffusion introduces further instability questions beyond this adiabatic parcel test.

For a grey atmosphere with $T^4=\frac{3}{4}{T}_{\mathrm{eff}}^4(\tau +2/3)$ and $P^2=P_0^2\log (1+3\tau /2)$, logarithmic differentiation givesA monatomic perfect gas of uniform composition has adiabatic temperature gradient $2/5$. The Schwarzschild criterion therefore predicts the first marginally unstable layer at ${\tau }_c=\frac{2}{3}(e^{4/5}-1)$, and convection for larger optical depth. Beyond that onset, the assumed purely radiative profile is no longer a self-consistent transport solution.

A convective core is a central stellar region in which the radiative temperature gradient exceeds the adiabatic threshold. Efficient convection makes its entropy nearly uniform and mixes its composition.

The global Eddington closure $L=(1-\beta )4\pi cGM/\kappa$, combined with a representative stellar gas-pressure fraction, makes the exterior stellar radiative temperature gradient approximately $M/(4m)$ when the luminosity is constant outside the core. Matching it to the adiabatic temperature gradient estimates the displayed boundary mass. This is a closure estimate: imposing exactly uniform $\beta$ throughout a finite-mass radiative envelope also conflicts with constant luminosity, as described by the constant-beta radiative-envelope obstruction.

A convective envelope is an outer stellar region in which convection carries a substantial fraction of the energy and mixes material. Its nearly adiabatic response to rapid mass loss is important for the dynamical stability of binary mass transfer.

A radiative envelope transports most of its energy by radiative diffusion. Its stratified entropy profile gives a different rapid mass-loss response from a convective envelope.

For a negligible-mass ideal-gas envelope above a core of mass $M_c$, with constant luminosity $L$ and Kramers' opacity law, division of hydrostatic equilibrium by radiative diffusion gives $PdP/dT=16\pi acG{M}_c\mathcal{R}{T}^{15/2}/(3{\kappa }_{0}L\mu )$. Taking the outer pressure constant to be negligible yields $P=C{T}^{17/4}$, where $C^2=64\pi acG{M}_c\mathcal{R}/(51{\kappa }_{0}L\mu )$, and $\rho =(\mu C/\mathcal{R})T^{13/4}$. Hydrostatic equilibrium then gives $dT/dr=-4\mu G{M}_c/(17\mathcal{R}{r}^2)$, so $T=B/r+T_0$ with $B=4\mu G{M}_c/(17\mathcal{R})$. The radiative-zero choice $T_0=0$ is exact for a vanishing-temperature boundary at infinity and a deep-envelope approximation for a very extended finite star. It is not implied by small surface temperature alone at a specified finite radius.

For a constant-luminosity radiative envelope with ideal-gas pressure and Kramers' opacity law, impose power laws on enclosed mass, density, pressure and temperature while retaining $M_r^{\prime }=4\pi r^2\rho$. If their exponents are $m,b,p,d$, respectively, mass conservation gives $m=b+3$, hydrostatic balance gives $d=m-1$, the gas law gives $d=p-b$, and diffusion gives $d-1=2b-2-13d/2$. Solving gives $(m,b,p,d)=(1,-32,-42,-10)/11$. In particular $R/r_c=(T_c/T_R{)}^{11/10}$. Neglecting envelope self-gravity instead sets $M_r$ approximately constant and changes the exponents to $d=-1,b=-13/4,p=-17/4$; that approximation is not an exact solution of mass conservation with nonzero density.

A thin radiative envelope with constant enclosed mass, luminosity and mean molecular weight, ideal gas pressure and opacity $\kappa ={\kappa }_0{\rho }^n{T}^{-s}$ has $a=n+1$, $b=n+s+4$, and $C=16\pi a_{\mathrm{rad}}cGM/[3{\kappa }_0(\mu m_u/k_B{)}^{n}L]$. Combining radiative diffusion in a star with hydrostatic equilibrium gives ${\nabla }_{\mathrm{rad}}=P^a/(C{T}^b)$, so $P^a-P_0^a=(aC/b)(T^b-T_0^b)$ for nonzero exponents. Zero exponents give logarithmic limits. The model must be stopped or modified if the Schwarzschild criterion predicts convection.

For $P=C(T^b+K{)}^{1/p}$ with $p=n+1$ and $b=4+n+m$, the logarithmic radiative temperature gradient is $(p/b)(1+K/T^b)$. At a marginal boundary with an efficient ideal-gas convective envelope, the Schwarzschild criterion equates it to $(\gamma -1)/\gamma$. Hence $K=T_b^b[(b/p)(\gamma -1)/\gamma -1]$. The notation $K=T_0^b$ with real positive $T_0$ requires a nonnegative bracket. Vanishing $p$ or $b$ needs the corresponding logarithmic integration instead of this power form.

In a negligible-mass, non-burning stellar envelope with $P=K{T}^{5/2}$ and Kramers' opacity law $\kappa ={\kappa }_1\rho T^{-7/2}$, the stellar radiative temperature gradient is ${\nabla }_{\mathrm{rad}}=3{\kappa }_1\mu K^{2}L/(16\pi ac\mathcal{R}GM{T}^{7/2})$. It decreases inward as the temperature rises, so the inner edge of efficient convection is where it equals $2/5$. If outer matching gives $K\propto g^{1/2}{T}_e^{-2\beta -5/2}$, then $T_b^{7/2}\propto K^{2}L/M\propto T_e^{-4\beta -1}$, since $g=GM/R^2$ and $L=4\pi R^2\sigma T_e^4$. Hence $T_b\propto T_e^{-(8\beta +2)/7}$.

A fully convective star has convection throughout nearly its entire interior. Very low-mass red dwarfs are fully convective; the transition to this structure is used in the disrupted magnetic braking model.